Defeated Sen. Scott Brown’s GOP colleagues are already recruiting him to run for John Kerry’s Senate seat.

Sen. Dan Coats (R-Ind.) personally implored Brown this month to run again. Sen. John McCain (R-Ariz.), a mentor who serves with Brown on the Armed Services Committee, will soon make a similar pitch. And retiring Sen. Olympia Snowe (R-Maine) has asked him about a possible return, saying Brown’s moderate voice and record of bipartisanship are sorely needed in the gridlocked chamber.

“I didn’t get a definite answer,” Coats said Friday. “He said, ‘Well, thanks for the advice.’”

President Barack Obama’s official nomination of Kerry as secretary of state triggered a flurry of speculation and behind-the-scenes jockeying over who might replace the senior senator from Massachusetts, who’s almost certain to receive a smooth confirmation from his fellow senators.

A poll this week showed Brown would beat top Bay State Democrats including Gov. Deval Patrick and Brown’s 2010 special election foe, Attorney General Martha Coakley, if the election were held today. Patrick has already said he will not run.

The Democratic primary could be a free-for-all. Reps. Michael Capuano, Stephen Lynch and Ed Markey are all believed to be mulling a run, though Markey — the dean of the Massachusetts delegation — may be the only one who makes a serious bid, sources said. Lynch is against abortion, which could complicate his chances.

Others floated include actor and Massachusetts native Ben Affleck and lesser-known names like state Sen. Ben Downing and former Taunton City Councilor Dave Simas, who worked as a policy adviser in the Obama White House.

“Congressman Capuano is looking forward to spending the holidays with his family and talking with them about his future,” said his district director, Jon Lenicheck. “Sen. John Kerry has served the commonwealth with great distinction and the congressman knows he will be an excellent secretary of state.”

Kennedy allies have been talking up Ted Kennedy Jr. as a possible successor; his father served alongside Kerry from 1985 until his death in 2009. The younger Kennedy was on hand for the failed Senate vote earlier this month of the United Nations disability treaty, a cause championed by Kerry. But Kennedy Jr., a lawyer and investment banker, currently lives in Connecticut and would have to move.

“I am unsure if he could clear the field, and there may be more than a little angst” over his residency, said one Massachusetts Democrat who’s run several races in the state.

Under state elections law, Patrick will have to appoint a temporary successor when Kerry is confirmed, just as former Democratic National Committee Chairman Paul Kirk was tapped to replace Kennedy Sr. upon his death. Kennedy’s widow, Vicki Kennedy, has been mentioned as a possible interim appointment for the Kerry seat, as have retiring Rep. Barney Frank and former Democratic presidential nominee Michael Dukakis.

A special general election will be held between 145 and 160 days from the time of the vacancy, preceded by a special primary. With such a short window, a successful candidate will need to have a strong fundraising network and name ID. That plays to Brown’s favor.

Brown’s coming off a tough defeat to Democratic consumer advocate Elizabeth Warren, and a 2013 special election would be his third statewide race in just four years. But his approval numbers are still high among Massachusetts voters — something that hasn’t gone unnoticed by his Senate colleagues.

“I would certainly encourage him. I think he’s done a great job in the Senate and I noticed that even though he lost his election against now-Sen. Warren, his favorables were very high, so I hope he will,” said McCain, who presided over a ceremony this year promoting Brown to the rank of colonel in the Army National Guard. “But those are decisions only he and his family can make.”

Brown had no comment about whether he was interested in running for the Kerry seat, his spokeswoman said. His father died on Thursday and he flew home to be with family on Friday, missing Senate votes on the defense authorization bill and Hurricane Sandy aid.

But Brown understands the politics of deep-blue Massachusetts, having won in 2010 the seat long held by the late liberal lion of the Senate, Ted Kennedy. Brown has often crossed the aisle on key votes and touted his work with Obama — he was even the first Republican to call for the assault weapons ban to be renewed following the deadly school shooting in Newtown, Conn.

“I’d love to have him run for the Senate. He’s a great voice. We need that bipartisanship,” said Snowe, who is leaving the Senate out of frustration it’s become too partisan. “Not only is he a true believer, but he’s also lived the bipartisanship. He has played that role significantly, and obviously he would be a crucial player in the United States Senate, so hopefully he moves in that direction.”